Prime Numbers

478 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
that some of them are also suited for the goal of convolution, so we name a
few: The Walsh–Hadamard transform, for which one needs no multiplication,
only addition; the discrete cosine transform (DCT), which is a real-signal,
real-multiplication analogue to the DFT; various wavelet transforms, which
sometimes admit of very fast (O(N) rather than O(N ln N)) algorithms; realvalued
FFT, which uses either cos or sin in real-only summands; the realsignal
Hartley transform, and so on. Various of these options are discussed in
[Crandall 1994b, 1996a].
Just to clear the air, we hereby make explicit the almost trivial difference
between the DFT and the celebrated fast Fourier transform (FFT). The
FFT is an operation belonging to the general class of divide-and-conquer
algorithms, and which calculates the DFT of Definition 9.5.3. The FFT will
typically appear in our algorithm layouts in the form X = FFT(x), where
it is understood that the DFT is being calculated. Similarly, an operation
FFT −1 (x) returns the inverse DFT. We make the distinction explicit because
“FFT” is in some sense a misnomer: The DFT is a certain sum—an algebraic
quantity—yet the FFT is an algorithm. Here is a heuristic analogy to the
distinction: In this book, the equivalence class x (mod N) are theoretical
entities, whereas the operation of reducing x modulo p we have chosen to
write a little differently, as x mod p. Bythesametoken,withinanalgorithm
the notation X = FFT(x) means that we are performing an FFT operation
on the signal X; and this operation gives, of course, the result DFT(x). (Yet
another reason to make the almost trivial distinction is that we have known
students who incorrectly infer that an FFT is some kind of “approximation”
to the DFT, when in fact, the FFT is sometimes more accurate then a literal
DFT summation, in the sense of roundoff error, mainly because of reduced
operation count for the FFT.)
The basic FFT algorithm notion has been traced all the way back to
some observations of Gauss, yet some authors ascribe the birth of the modern
theory to the Danielson–Lanczos identity, applicable when the signal length
D is even:
D−1
DFT(x) = xjg
j=0
−jk D/2−1 2
= x2j g
j=0
D/2−1
−jk −k
2
+ g x2j+1 g
j=0
−jk .
(9.22)
A beautiful identity indeed: A DFT sum for signal length D is split into two
sums, each of length D/2. In this way the Danielson–Lanczos identity ignites
a recursive method for calculating the transform. Note the so-called twiddle
factors g−k , which figure naturally into the following recursive form of FFT.
In this and subsequent algorithm layouts we denote by len(x) the length of
asignalx. In addition, when we perform element concatenations of the form
(aj)j∈J we mean the result to be a natural, left-to-right, element concatenation
as the increasing index j runs through a given set J. Similarly, U ∪ V is a
signal having the elements of V appended to the right of the elements of U.